Recent questions tagged tifr2014 / GO Electronics

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TIFR ECE 2014 | Question: 1
Consider two independent and identically distributed random variables $X$ and $Y$ uniformly distributed in $[0,1]$. For $\alpha \in[0,1]$, the probability that $\alpha \max (X, Y)<\min (X, Y)$ is $1 /(2 \alpha)$. $\exp (1-\alpha)$ $1-\alpha$ $(1-\alpha)^{2}$ $1-\alpha^{2}$

Consider two independent and identically distributed random variables $X$ and $Y$ uniformly distributed in $[0,1]$. For $\alpha \in[0,1]$, the probability that $\alpha \m...

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2

TIFR ECE 2014 | Question: 2
Evaluate the limit \[ \lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} . \] $e$ $1$ $2^{\frac{1}{3}}$ $0$ None of the above

Evaluate the limit\[\lim _{n \rightarrow \infty}\left(2 n^{4}\right)^{\frac{1}{3 n}} .\]$e$$1$$2^{\frac{1}{3}}$$0$None of the above

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3

TIFR ECE 2014 | Question: 3
For a non-negative continuous random variable $X$, which of the following is TRUE? $E\{X\}=\int_{0}^{\infty} P(X>x) d x$. $E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$. $P(X<x) \leq \frac{E\{X\}}{x}$. $(a)$ and $(c)$. None of the above.

For a non-negative continuous random variable $X$, which of the following is TRUE?$E\{X\}=\int_{0}^{\infty} P(X>x) d x$.$E\{X\}=\int_{0}^{\infty} P(X \leq x) d x$.$P(X<x)...

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4

TIFR ECE 2014 | Question: 4
A system accepts a sequence of real numbers $x[n]$ as input and outputs \[ y[n]=\left\{\begin{array}{ll} 0.5 x[n]-0.25 x[n-1], & n \text { even } \\ 0.75 x[n], & n \text { odd } \end{array}\right. \] The system is non-linear. non-causal. time-invariant. All of the above. None of the above.

A system accepts a sequence of real numbers $x[n]$ as input and outputs\[y[n]=\left\{\begin{array}{ll}0.5 x[n]-0.25 x[n-1], & n \text { even } \\0.75 x[n], & n \text { od...

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5

TIFR ECE 2014 | Question: 5
The matrix \[ A=\left(\begin{array}{ccc} 1 & a_{1} & a_{1}^{2} \\ 1 & a_{2} & a_{2}^{2} \\ 1 & a_{3} & a_{3}^{2} \end{array}\right) \] is invertible when $a_{1}>a_{2}>a_{3}$ $a_{1}<a_{2}<a_{3}$ $a_{1}=3, a_{2}=2, a_{3}=4$ All of the above None of the above

The matrix\[A=\left(\begin{array}{ccc}1 & a_{1} & a_{1}^{2} \\1 & a_{2} & a_{2}^{2} \\1 & a_{3} & a_{3}^{2}\end{array}\right)\]is invertible when$a_{1}>a_{2}>a_{3}$$a_{1}...

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6

TIFR ECE 2014 | Question: 6
Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy \[ \int_{0}^{\pi} g(x) \sin (n x) d x=0 \] for all integers $n \geq 2$. Then which of the following can you say about $g?$ $g$ must be identically zero. $g(\pi / 2)=1$. $g$ need not be identically zero. $g(\pi)=0$. None of the above.

Let $g:[0, \pi] \rightarrow \mathbb{R}$ be continuous and satisfy\[\int_{0}^{\pi} g(x) \sin (n x) d x=0\]for all integers $n \geq 2$. Then which of the following can you ...

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7

TIFR ECE 2014 | Question: 7
Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of the following can we conclude about $A?$ All eigenvalues of $A$ are non-negative. $A$ is not full rank. $A$ is not the zero matrix. $\operatorname{det}(A) \neq 0$. None of the above.

Let $A$ be an $n \times n$ real matrix. It is known that there are two distinct $n$-dimensional real column vectors $v_{1}, v_{2}$ such that $A v_{1}=A v_{2}$. Which of t...

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8

TIFR ECE 2014 | Question: 8
Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fold convolution. Let $f(t)=\lim _{n \rightarrow \infty} f_{n}(t)$. Then, which ... $\infty$. $f(t)$ has width $\infty$ and height $1$ . $f(t)$ has width $0$ and height $\infty$. None of the above.

Consider a square pulse $g(t)$ of height $1$ and width $1$ centred at $1 / 2$. Define $f_{n}(t)=\frac{1}{n}\left(g(t) *^{n} g(t)\right),$ where $*^{n}$ stands for $n$-fol...

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9

TIFR ECE 2014 | Question: 9
Consider the following input $x(t)$ and output $y(t)$ pairs for two different systems. $x(t)=\sin (t), y(t)=\cos (t),$ $x(t)=t+\sin (t), y(t)=2 t+\sin (t-1).$ Which of these systems could possibly be linear and time invariant? Choose the most appropriate answer ... i) nor (ii). neither, but a system with $x(t)=\sin (2 t), y(t)=\sin (t) \cos (t) \operatorname{could~be.~}$

Consider the following input $x(t)$ and output $y(t)$ pairs for two different systems.$x(t)=\sin (t), y(t)=\cos (t),$$x(t)=t+\sin (t), y(t)=2 t+\sin (t-1).$Which of these...

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10

TIFR ECE 2014 | Question: 10
Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbol interference. The constellation points are picked equally likely. Let $P\text{(QAM)}$ denote the ... .

Consider the two quadrature amplitude modulation $\text{(QAM)}$ constellations below. Suppose that the channel has additive white Gaussian noise channel and no intersymbo...

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11

TIFR ECE 2014 | Question: 11
It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class: \[ \left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq A \leq 1,0<\theta \leq \pi\right\} \] If you ... how many samples are required to determine the signal? $1$ sample. $2$ samples. $1$ sample per second. $2$ samples per second. None of the above.

It is known that the signal $x(t)$, where $t$ denotes time, belongs to the following class:\[\left\{A \sin \left(2 \pi f_{0} t+\theta\right): f_{0}=1 \mathrm{~Hz}, 0 \leq...

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12

TIFR ECE 2014 | Question: 12
Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal value of $\alpha$ which minimizes $\mathbb{E}\left[(X-\alpha Y)^{2}\right]$ ... $1$ $\frac{\sigma_{Y}^{2}}{\sigma_{Z}^{2}}$ None of the above.

Assume that $Y, Z$ are independent, zero-mean, continuous random variables with variances $\sigma_{Y}^{2}$ and $\sigma_{Z}^{2},$ respectively. Let $X=Y+Z$. The optimal va...

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13

TIFR ECE 2014 | Question: 13
Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y)$. Then which of the following is $\text{TRUE?}$ $f(x) \leq f(y)$ whenever ... $f$ and $g$ are both convex, then $\min \{f, g\}$ is also convex. For a random variable $X, E(f(X)) \geq f(E(X))$.

Let function $f: \mathbf{R} \rightarrow \mathbf{R}$ be convex, i.e., for $x, y \in \mathbf{R}, \alpha \in[0,1], f(\alpha x+(1-\alpha) y) \leq$ $\alpha f(x)+(1-\alpha) f(y...

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14

TIFR ECE 2014 | Question: 14
Suppose that a random variable $X$ has a probability density function \[ \begin{aligned} f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\ & =0 \quad \text { for all other } x \end{aligned} \] for some constant $c$. What is the expected value of $X$ given that $X \geq 5?$ $5 \frac{5}{9}$ $5 \frac{1}{2}$ $5 \frac{3}{4}$ $5 \frac{1}{4}$ $5 \frac{5}{8}$

Suppose that a random variable $X$ has a probability density function\[\begin{aligned}f(x) & =c(x-4) \quad \text { for } 4 \leq x \leq 6 \\& =0 \quad \text { for all othe...

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15

TIFR ECE 2014 | Question: 15
You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs. $10$ per meter and the wall paper can be put at the rate of Rs $20$ per meter for that ... $200$ square meter room? $400 \times \sqrt{3} $ $400$ $400 \times \sqrt{2}$ $200 \times \sqrt{3}$ $500$

You are allotted a rectangular room of a fixed height. You have decided to paint the three walls and put wallpaper on the fourth one. Walls can be painted at a cost of Rs...

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16

TIFR ECE 2014 | Question: 16
A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is $4 \frac{1}{2}$ $3 \frac{1}{2}$ $5$ $4 \frac{17}{36} $ $4 \frac{3}{4}$

A fair dice (with faces numbered $1, \ldots, 6$ ) is independently rolled twice. Let $X$ denote the maximum of the two outcomes. The expected value of $X$ is$4 \frac{1}{2...

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17

TIFR ECE 2014 | Question: 17
Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian distributed with mean $\mu_{2}$ and variance $\sigma_{2}^{2}$. Then the distribution ... variable with mean $\mu_{2}$ and variance $\sigma_{1}^{2}+\sigma_{2}^{2}$. Has no known form. None of the above.

Let $X$ be a Gaussian random variable with mean $\mu_{1}$ and variance $\sigma_{1}^{2}$. Now, suppose that $\mu_{1}$ itself is a random variable, which is also Gaussian d...

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18

TIFR ECE 2014 | Question: 18
A non-negative loss in a car accident is distributed with the following probability density function \[ f(x)=\frac{1}{10} \exp (-x / 10) \] for $x \geq 0$. Suppose that first $5$ units of loss is incurred by the insured and the remaining loss if any is covered by the ... $5+10 \exp \left(-\frac{1}{2}\right)$ $15 \exp \left(-\frac{1}{2}\right)$

A non-negative loss in a car accident is distributed with the following probability density function\[f(x)=\frac{1}{10} \exp (-x / 10)\]for $x \geq 0$. Suppose that first...

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19

TIFR ECE 2014 | Question: 19
Consider a $2^{k} \times N$ binary matrix $A=\left\{a_{\ell, k}\right\}, a_{\ell, k} \in\{0,1\}$. For rows $i$ and $j$, let the Hamming distance be $d_{i, j}=\sum_{\ell=1}^{N}\left|a_{i, \ell}-a_{j, \ell}\right|$. Let $D_{\min }=\min _{i, j} d_{i, j}$. ... $D_{\min } \leq N-k+1$. $D_{\min } \leq N-k$. $D_{\min } \leq N-k-1$. $D_{\min } \leq N-k-2$. None of the above.

Consider a $2^{k} \times N$ binary matrix $A=\left\{a_{\ell, k}\right\}, a_{\ell, k} \in\{0,1\}$. For rows $i$ and $j$, let the Hamming distance be $d_{i, j}=\sum_{\ell=1...

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20

TIFR ECE 2014 | Question: 20
What is \[ \lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ? \] $0$ $\pi / 2$ $1 / \sqrt{2}$ $2 / \pi$ None of the above.

What is\[\lim _{n \rightarrow \infty} \cos \frac{\pi}{2^{2}} \cos \frac{\pi}{2^{3}} \cdots \cos \frac{\pi}{2^{n}} ?\]$0$$\pi / 2$$1 / \sqrt{2}$$2 / \pi$None of the above....

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